### 3.979 $$\int (d+e x) (c d^2+2 c d e x+c e^2 x^2) \, dx$$

Optimal. Leaf size=15 $\frac{c (d+e x)^4}{4 e}$

[Out]

(c*(d + e*x)^4)/(4*e)

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Rubi [A]  time = 0.0040956, antiderivative size = 15, normalized size of antiderivative = 1., number of steps used = 3, number of rules used = 3, integrand size = 26, $$\frac{\text{number of rules}}{\text{integrand size}}$$ = 0.115, Rules used = {27, 12, 32} $\frac{c (d+e x)^4}{4 e}$

Antiderivative was successfully veriﬁed.

[In]

Int[(d + e*x)*(c*d^2 + 2*c*d*e*x + c*e^2*x^2),x]

[Out]

(c*(d + e*x)^4)/(4*e)

Rule 27

Int[(u_.)*((a_) + (b_.)*(x_) + (c_.)*(x_)^2)^(p_.), x_Symbol] :> Int[u*Cancel[(b/2 + c*x)^(2*p)/c^p], x] /; Fr
eeQ[{a, b, c}, x] && EqQ[b^2 - 4*a*c, 0] && IntegerQ[p]

Rule 12

Int[(a_)*(u_), x_Symbol] :> Dist[a, Int[u, x], x] /; FreeQ[a, x] &&  !MatchQ[u, (b_)*(v_) /; FreeQ[b, x]]

Rule 32

Int[((a_.) + (b_.)*(x_))^(m_), x_Symbol] :> Simp[(a + b*x)^(m + 1)/(b*(m + 1)), x] /; FreeQ[{a, b, m}, x] && N
eQ[m, -1]

Rubi steps

\begin{align*} \int (d+e x) \left (c d^2+2 c d e x+c e^2 x^2\right ) \, dx &=\int c (d+e x)^3 \, dx\\ &=c \int (d+e x)^3 \, dx\\ &=\frac{c (d+e x)^4}{4 e}\\ \end{align*}

Mathematica [A]  time = 0.0016316, size = 15, normalized size = 1. $\frac{c (d+e x)^4}{4 e}$

Antiderivative was successfully veriﬁed.

[In]

Integrate[(d + e*x)*(c*d^2 + 2*c*d*e*x + c*e^2*x^2),x]

[Out]

(c*(d + e*x)^4)/(4*e)

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Maple [B]  time = 0.039, size = 36, normalized size = 2.4 \begin{align*}{\frac{c{e}^{3}{x}^{4}}{4}}+dc{e}^{2}{x}^{3}+{\frac{3\,{d}^{2}ec{x}^{2}}{2}}+c{d}^{3}x \end{align*}

Veriﬁcation of antiderivative is not currently implemented for this CAS.

[In]

int((e*x+d)*(c*e^2*x^2+2*c*d*e*x+c*d^2),x)

[Out]

1/4*c*e^3*x^4+d*c*e^2*x^3+3/2*d^2*e*c*x^2+c*d^3*x

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Maxima [B]  time = 1.1008, size = 41, normalized size = 2.73 \begin{align*} \frac{{\left (c e^{2} x^{2} + 2 \, c d e x + c d^{2}\right )}^{2}}{4 \, c e} \end{align*}

Veriﬁcation of antiderivative is not currently implemented for this CAS.

[In]

integrate((e*x+d)*(c*e^2*x^2+2*c*d*e*x+c*d^2),x, algorithm="maxima")

[Out]

1/4*(c*e^2*x^2 + 2*c*d*e*x + c*d^2)^2/(c*e)

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Fricas [B]  time = 1.56161, size = 77, normalized size = 5.13 \begin{align*} \frac{1}{4} x^{4} e^{3} c + x^{3} e^{2} d c + \frac{3}{2} x^{2} e d^{2} c + x d^{3} c \end{align*}

Veriﬁcation of antiderivative is not currently implemented for this CAS.

[In]

integrate((e*x+d)*(c*e^2*x^2+2*c*d*e*x+c*d^2),x, algorithm="fricas")

[Out]

1/4*x^4*e^3*c + x^3*e^2*d*c + 3/2*x^2*e*d^2*c + x*d^3*c

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Sympy [B]  time = 0.075352, size = 39, normalized size = 2.6 \begin{align*} c d^{3} x + \frac{3 c d^{2} e x^{2}}{2} + c d e^{2} x^{3} + \frac{c e^{3} x^{4}}{4} \end{align*}

Veriﬁcation of antiderivative is not currently implemented for this CAS.

[In]

integrate((e*x+d)*(c*e**2*x**2+2*c*d*e*x+c*d**2),x)

[Out]

c*d**3*x + 3*c*d**2*e*x**2/2 + c*d*e**2*x**3 + c*e**3*x**4/4

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Giac [B]  time = 1.18386, size = 46, normalized size = 3.07 \begin{align*} \frac{1}{4} \, c x^{4} e^{3} + c d x^{3} e^{2} + \frac{3}{2} \, c d^{2} x^{2} e + c d^{3} x \end{align*}

Veriﬁcation of antiderivative is not currently implemented for this CAS.

[In]

integrate((e*x+d)*(c*e^2*x^2+2*c*d*e*x+c*d^2),x, algorithm="giac")

[Out]

1/4*c*x^4*e^3 + c*d*x^3*e^2 + 3/2*c*d^2*x^2*e + c*d^3*x